arxiv: 2605.03535 · v1 · submitted 2026-05-05 · 💻 cs.FL

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Hyper-Minimization for Deterministic Register Automata

Yong Li , Qiyi Tang , Di-De Yen

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Pith reviewed 2026-05-09 15:58 UTC · model grok-4.3

classification 💻 cs.FL

keywords deterministic register automatahyper-minimizationwell-typed DRAsdecidabilitystate minimizationregister minimizationfinite automataautomata theory

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The pith

Well-typed deterministic register automata can be hyper-minimized to the smallest number of states and registers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts standard minimization ideas from ordinary finite automata to deterministic register automata that store and compare data values in registers. It restricts attention to well-typed DRAs, in which each state is tied to exactly one register type that dictates the kind of data the state can process. An algorithm is supplied that rewrites any such automaton into an equivalent one that uses fewer states and fewer registers. The authors show that the algorithm is correct and always terminates, which immediately makes the question of whether a given well-typed DRA is already minimal decidable. This matters for any verification or analysis task that works with data-aware automata, because smaller models are cheaper to store and check.

Core claim

We present an algorithm for hyper-minimizing well-typed DRAs, where each state is associated with a unique register type. The resulting automata are minimal with respect to both the number of states and registers among all well-typed DRAs. We prove the correctness of the proposed algorithm, thereby establishing the decidability of hyper-minimization for well-typed DRAs.

What carries the argument

The hyper-minimization algorithm for well-typed DRAs that produces an equivalent automaton minimal in both states and registers by using the unique register type per state.

Load-bearing premise

The algorithm and the decidability result apply only when the DRA is well-typed, so that every state has exactly one associated register type.

What would settle it

A concrete well-typed DRA for which the algorithm returns an automaton that still has more states or registers than some other equivalent well-typed DRA would falsify the minimality claim.

Figures

Figures reproduced from arXiv: 2605.03535 by Di-De Yen, Qiyi Tang, Yong Li.

**Figure 1.** Figure 1: An example DRA A Example 2. Let Lmid = {a1a2a3 | a1 < a3 < a2} over (Q, <). The DRA in view at source ↗

**Figure 2.** Figure 2: A hyper-data-minimal DRA A recognizing Lmid_plus. E Myhill-Nerode Theorem for DRAs Definition 27 ( [4]). Given a data language L over (Σ, R), we define an equivalence relation ∼=L on Σ∗ , where for every u, v ∈ Σ∗ , we write u ∼=L v if: (1) memL(u) ∼R memL(v), and (2) for all x, y ∈ Σ∗ , if memL(u) · x ∼R memL(v) · y, then u · x ∈ L ⇔ v · y ∈ L view at source ↗

read the original abstract

We investigate hyper-minimization for deterministic register automata (DRAs). We begin by introducing DRA counterparts of classical notions from deterministic finite automata. Building on these foundations, we present an algorithm for hyper-minimizing well-typed DRAs, where each state is associated with a unique register type. The resulting automata are minimal with respect to both the number of states and registers among all well-typed DRAs. We prove the correctness of the proposed algorithm, thereby establishing the decidability of hyper-minimization for well-typed DRAs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a hyper-minimization algorithm and correctness proof for well-typed DRAs where each state has a unique register type, settling decidability inside that narrow class.

read the letter

This paper supplies an algorithm that hyper-minimizes well-typed deterministic register automata and proves the result is minimal in both states and registers. It also shows the problem is decidable for exactly this subclass. They first lift the usual DFA notions of equivalence and minimization to the DRA setting, then use those to drive the reduction while respecting the typing constraint on registers. That is the concrete new piece: a working procedure plus a proof that it produces the smallest automaton inside the allowed class. The restriction to unique register types per state is stated up front, so the claim stays honest about its reach. If the proof is complete, the work adds a usable tool for people who already work with typed register automata. The main limitation is the typing restriction itself. Many DRAs in the literature do not assign a single distinct type to each state, so the algorithm does not apply to them and the paper does not claim it does. Without seeing the full proof details it is hard to judge how cleanly the construction handles register updates on transitions or edge cases with empty registers, but the abstract gives no sign of circularity or unfalsifiable steps. The paper is aimed at automata theorists who care about register models and minimization questions. A reader already following work on DRAs or similar extensions of finite automata will find the result relevant and can check the proof directly. It is narrow enough that it will not move the broader field, but the algorithmic contribution and decidability result are clear enough to merit referee time rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper investigates hyper-minimization for deterministic register automata (DRAs). It introduces DRA counterparts of classical DFA notions such as equivalence and minimization, then presents an algorithm for hyper-minimizing well-typed DRAs in which each state is associated with a unique register type. The resulting automata are claimed to be minimal with respect to both the number of states and the number of registers among all well-typed DRAs. A correctness proof for the algorithm is provided, which establishes the decidability of hyper-minimization for this restricted class.

Significance. If the algorithm and its correctness proof hold, the result provides a decidable procedure for finding automata that are minimal in both states and registers within the class of well-typed DRAs with unique per-state register types. This extends classical minimization techniques to register automata, a model relevant to data words and XML processing. The explicit scoping to well-typed DRAs avoids overclaiming generality, and the algorithmic construction plus proof constitute a concrete contribution to automata minimization.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a brief high-level sketch of the algorithm (e.g., the main steps or data structures used) to give readers an immediate sense of the construction before the detailed presentation.
[Preliminaries] Notation for register types and well-typedness should be introduced with a small illustrative example early in the preliminaries to aid readability for readers unfamiliar with register automata.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our work on hyper-minimization for deterministic register automata. The recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces DRA counterparts of classical DFA notions, then gives an explicit algorithm for hyper-minimizing the restricted class of well-typed DRAs (unique register type per state) and proves its correctness, thereby establishing decidability inside that class. No equations, fitted parameters, or predictions appear; the argument is a standard constructive proof with an upfront scope restriction and does not reduce any load-bearing step to a self-definition, self-citation chain, or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, invented entities, or non-standard axioms are stated. The work builds on classical DFA notions.

axioms (1)

domain assumption Standard notions and properties of deterministic finite automata extend to DRAs
Abstract states that DRA counterparts of classical DFA notions are introduced.

pith-pipeline@v0.9.0 · 5374 in / 1062 out tokens · 54737 ms · 2026-05-09T15:58:37.014858+00:00 · methodology

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Works this paper leans on

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Given that the register type of the initial state isϵ, the definition of almost-equivalence over states implies that (qA 0 , ϵ)≈(q B 0 , ϵ)

Furthermore, becauseA ≈ B, we have qA 0 ≈q B 0 and, consequently,q A 0 ≈h(q A 0 ). Given that the register type of the initial state isϵ, the definition of almost-equivalence over states implies that (qA 0 , ϵ)≈(q B 0 , ϵ). Now, letp∈Q A andq∈Q B such thath(p) =q. By definition,(q A 0 , ϵ) wp − − →A (p, u)and(q B 0 , ϵ) wp − − →B (q, v)for someu, v. By Le...